1. Technical Field
The present invention relates to a method, computer system, and program product for optimizing a useful objective function with respect to a dynamic parameter vector and a static parameter vector, and more particularly, to optimizing a spacecraft objective such as a spacecraft trajectory objective and a spacecraft design objective.
2. Related Art
A goal of spacecraft trajectory optimization is to obtain the best trajectory. Spacecraft trajectory optimization is a dynamic problem. A dynamic problem involves a sequence of decisions made over time. For example, the thrust direction for a spacecraft engine must be selected on day one of a space flight. Then a new thrust direction must be selected for day two, and so on. Each previous decision affects all future decisions. A complete set of decisions for the total flight time is necessary to define a trajectory.
A goal of spacecraft design optimization is to obtain the best design. Spacecraft design optimization is a not a dynamic problem. Design decisions are not linked to a time sequence. For example, the decisions of how large a solar array to build, how much fuel to carry, and when to launch a spacecraft are static or non-dynamic decisions.
The overall performance of a spacecraft is a function of both the spacecraft trajectory and the spacecraft design. Further, the spacecraft design will impact the spacecraft trajectory and vice versa. Since spacecraft design and spacecraft trajectory are not independent, the most efficient spacecraft design and the most efficient spacecraft trajectory cannot be determined independently.
Current methods for optimization are constructed to solve either dynamic problems or static problems. Related art optimization methods cannot solve problems which have both a static part and a dynamic part without approximation. Related art methods for optimization can be divided into two categories. The first category is non-dynamic or static optimization methods known as xe2x80x9cparameter optimizationxe2x80x9d. The second category is dynamic optimization methods known as xe2x80x9coptimal control.xe2x80x9d
Parameter optimization was originally developed to solve static problems. Parameter optimization is directly applicable to static problems, such as spacecraft design. However, parameter optimization is not easily or accurately applicable to dynamic problems such as space flight trajectories. Parameter optimization can only be applied to dynamic problems when the true dynamic nature of the problem is approximated or removed entirely. Since parameter optimization cannot solve general dynamic problems without approximation, parameter optimization cannot solve the combined static and dynamic problem of spacecraft design and trajectory without approximation.
Optimal control was developed for exclusively dynamic problems. Optimal control is applicable to dynamic problems such as spacecraft trajectory optimization. However, optimal control is not suited to static problems such as spacecraft design.
The related art uses either parameter optimization or optimal control to design deep space missions. The related art cannot achieve the combined benefits of both parameter optimization and optimal control simultaneously. The following two subsections describe the parameter optimization and the optimal control methods presently in use.
The related art applies parameter optimization to the dynamic problem of spacecraft trajectory optimization by making several significant approximations. The physics of the trajectory problem is approximated by instantaneous impulses followed by ballistic coasts. The instantaneous impulses are intended to represent the effect of thrust on a spacecraft. A sequence of impulses and coasts can roughly approximate a continuously running engine. The ballistic coasts usually only account for the gravitational influence of a single central object (usually the sun.) All long range planetary gravitational effects are ignored. Since planetary gravitation is ignored, the ballistic coasts are perfect conic sections whose shape is always time independent. The time independence of the shape is a necessary assumption for the non-dynamic parameter optimization method.
Of particular interest is the ion propulsion engine. Ion propulsion engines are far more efficient than ordinary traditional chemical engines. Ion propulsion engines are expected to replace many applications of chemical engines in the near future. Ion propulsion engines have significantly different operating characteristics than chemical engines. In particular, ion engines typically operate continuously for days or even years at low thrust intensities. The impulse/ballistic coast approximation required by parameter optimization is a particularly poor approximation when applied to ion engines.
The parameter optimization method used in the related art requires prespecification of the sequence of planets that the spacecraft will pass by closely (xe2x80x9cplanetary flyby sequencexe2x80x9d). Thus, the planetary flyby sequence is not susceptible to parameter optimization. Individual planetary flybys are prespecified by fixed constraints. Prespecification of flybys greatly reduces the likelihood of discovering the most efficient flyby sequence.
Parameter optimization typically requires the physics of planetary flybys to be approximated as a collision at a single point in space. The spacecraft trajectory is propagated to the center of the flyby body without accounting for the gravity of the body. Flybys are then modeled as an instantaneous change in velocity without a change in the spacecraft position. With parameter optimization, the true extended spatial and dynamic nature of flybys is not correctly represented. This approximation significantly reduces the precision of parameter optimization solutions.
Related art parameter optimization methods do not optimize spacecraft thrust sequences directly. Instead, related art methods optimize instantaneous changes to spacecraft velocity. Optimizing velocity changes neglects or only approximates the effect of the dynamic nature of the spacecraft mass. The spacecraft mass is a decreasing function of time because fuel is burned. Neglecting this fact reduces the precision of parameter optimization solutions.
The related art uses an optimal control method to calculate or fine tune navigation paths or trajectories for spacecraft. The optimal control method is known as the calculus of variations (xe2x80x9cCOV.xe2x80x9d) The COV method is not capable of calculating optimal trajectories for spacecraft from scratch. An efficient trajectory and/or a prespecified flyby sequence must be supplied as input. The input trajectory, or trajectory associated with a prescribed flyby sequence, is not derived from a precise optimization procedure and is therefore not necessarily optimal. Similarly, the prespecified flyby sequence is not necessarily optimal.
Related art COV methods typically make several approximations similar to the approximations made by related art parameter optimization methods. For example, thrust is approximated as a sequence of impulses, spacecraft propagation only takes into account the Sun""s gravity, and planetary flybys are treated as an instantaneous change in spacecraft velocity. As a result, the COV method is limited in precision in the same way that the parameter optimization method is limited in precision.
The COV method also suffers from extreme sensitivity to flyby parameters. The extreme sensitivity of COV results in a substantial reduction in the improvement that COV can potentially achieve when flybys are involved. The sensitivity problem limits the COV method to consider trajectories with only a small number of flybys.
The main advantage of the COV method is that it is a dynamic method. The dynamic aspect of the COV method could, in theory, permit the correct representation of the dynamics of space flight. Unfortunately, the COV method is not robust enough to solve the trajectory problem without relying on significant approximations or requiring the prespecification of the flyby sequence.
Application of the preexisting general optimization methods, known as neural networks, genetic algorithms, and simulated annealing to static/dynamic problems, may produce inferior results that do not converge to optimal solutions. These optimization methods are heuristic or approximate methods involving various types of random searches to find good solutions. Heuristic methods often fail to find the best or optimal solutions.
An accurate method is needed to simultaneously optimize both dynamic and static variables collaboratively, such as by simultaneously taking into account the effect of navigation on spacecraft design, and vice-versa.
The present invention of static/dynamic control (xe2x80x9cSDCxe2x80x9d) provides a method for optimizing a useful objective, comprising the steps of:
providing a computer system having:
a memory device;
a code located on the memory device;
a processor for executing the code; and
a useful objective function J coupled to the code, wherein the code includes a static/dynamic control program for optimizing J with respect to a dynamic control vector v(t) and a static parameter vector w, and wherein t denotes time;
providing input data for the static/dynamic control program, wherein the input data is coupled to the code;
optimizing J with respect to v(t) and w, by executing the static/dynamic control program and using the input data; and
outputting a computed result from the optimizing step to an output device within the computer system.
The present invention provides a first computer system for optimizing a useful objective, comprising:
a memory device,
a computer code located on the memory device;
a computer processor for executing the computer code;
a useful objective function J coupled to the computer code, wherein the computer code includes a static/dynamic control program for performing an optimization of J with respect to a dynamic control vector v(t) and a static parameter vector w, and wherein t denotes time; input data for the static/dynamic control program; and an output device for receiving a result from the static/dynamic control program.
The present invention provides a second computer system for optimizing a useful objective, comprising:
a memory means for storing data on a memory device;
a computer code, located on the memory device, for optimizing a useful objective function J with respect to a dynamic control vector v(t) and a static parameter vector w, wherein t denotes time;
a processor means for executing the computer code;
an input means for transferring input data to the static/dynamic control program; and
an output means for receiving a result from the static/dynamic control program.
The present invention provides a computer program product, comprising:
a recordable medium; and
a computer code recorded on the recordable medium. wherein the computer code includes a static/dynamic control program.
By optimizing a useful objective, the present invention produces a useful, concrete, and tangible result. In particular, the useful objective may include a spacecraft objective, a groundwater decontamination objective, or a chemical reactor objective.
The present invention has the advantage of solving problems having both a static and a dynamic aspect. In the related art, optimization algorithms are designed for either static problems or dynamic problems, but not both.
The present invention has the advantage of solving a generalized state equation. The state equation include such equations as Newton""s equations of motion, general relativistic equations of motion, Darcy""s law of potential flow for groundwater movement, computational fluid dynamics equations for air flow over an aircraft, and transient equations for the transport of neutrons, radiation, heat, electric current, etc.
The present invention has the advantage of accommodating generalized physical constraints such as requiring: a minimum distance between a spacecraft and the sun so as to protect the spacecraft from overheating, a maximum feasible thrust magnitude for a spacecraft, or a maximum allowed temperature during operation of a chemical reactor in a chemical processing system.
The present invention has the advantage of allowing the initial or the terminal condition of the state of the system to be a variable. Related art optimal control methods require the initial state of the system to be given and fixed.
The present invention has the advantage of including an algorithm having two formulations: a period formulation and a fully continuous formulation. With the period formulation, the dynamic control is permitted to be a discontinuous, parametric function of time. The usefulness of the period formulation is illustrated by the capability of changing engine throttle once every fixed number of days, such as once per ten days. With the fully continuous formulation, the dynamic control is permitted to be non-parametric and continuous in time. Illustrative of the usefulness of the fully continuous formulation is the capability of satisfying a generally recognized need to optimize solar sail and light propulsion spacecraft, which are expected to be deployed in the next few years.
The present invention has the advantage of solving spacecraft optimization problems without approximation to the physics or dynamics involved. Among the effects susceptible of inclusion by SDC are general relativistic corrections, radiation pressure, continuous thrust, planetary harmonics, dynamic interaction between spacecraft and planetary flybys, gravitational influence of any number of solar system objects, and planetary drag on spacecraft traversing a planetary atmosphere. Related art methods require simplifying assumptions for compatibility with related art algorithms.
The present invention has the advantage of satisfying a generally recognized and existing need to optimize both ion propulsion spacecraft and combination chemical and ion propulsion spacecraft designs and trajectories. Related art methods perform particularly poorly for ion propulsion and general low thrust spacecraft.
The present invention has the advantage of enabling mathematical derivatives to be computed exactly. Exact derivatives result in a more robust process and faster convergence. The related art typically uses approximate derivatives.
The present invention has the advantage of employing a fully second order optimization algorithm which cannot converge to false minima or saddle points. Both the COV and parameter optimization methods currently in use can converge to false minima or saddle points.
The present invention has the advantage of converging only to optimal solutions which satisfy both the necessary and sufficient conditions of optimality. Existing heuristic optimization methods (neural networks, genetic algorithms, simulated annealing) often fail to converge to optimal solutions, since they are not designed to find optimal solutions.